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Unformatted text preview: UNl‘lFERSITY CIF CALIFORNIA, BERKELEY
Physics Department Th, Liphardt Fall Term 2005
Exam #1 Name
Disc. Section
SID #1
#2 #3
#4 Total ___ of too
The relative weight of each problem is indicated next to the problem number. Most credit will be given for algebraic work. Please do not insert numerical values until you
have a final algebraic answer inside a box. If you don’t ltnowr a particular constant, use a symbol
to get partial credit. It is almost impossible to award partial credit if you insert numbers too early.
If you get stuck on one problem, go on to the neat and come back to the difficulties later in the
exam period. Do not leave early until you have completed everything. Do not quit! Never, never,
never quitl! As usual, before you start to manipulate equations and numbers you should think for a moment
and make sure that you have a general sense of the important features of a problem. Also, once
you have computed something, check to see if your answer has the right units, and if it is in the
right ballpark. For example, if your answer to any problem is something like at a. 19““
Kelvini'meters, you should check your work. Finally, please do not violate the ﬁrst and second
laws! k3: 1.38x lﬂ'BJiK
t~~.1,,=sxtoE I'roblem 1 [25 points). A constant number of Nitrogen molecules are in a cylinder of variable
sise. Someone compresses the gas to _ of its original volume: Um = _ Vimml. Simultaneously,
energy is added to the gas by compression and heating, so that the temperature increases 5—fold:
Tm = 5 Tim“. Assume that the ideal gas law applies. By what factor do the following change? a) pressure b} kinetic energy c) rms speed of the molecules d) the number of impacts per second by molecules on 1 tom1 of wall area? Problem 2 (25 points]. Let’s ignite a fuel air mixture by compressing it (this is precisely what
happens in a diesel engine). Assume  slightly,r unrealistically — that the fuel—air mixture is an
ideal gas, and that the appropriate ratio of heat capacities is '3' = L4. Let’s compress the mixture
adiabatically from an initial volume of Vi to a ﬁnal volume of ﬂ.1*‘v’i. Initial T = 212] C. a} What is the ﬁnal temperature? in} By whatfocror does the pressure increase? It; Problem 3 (25 points). Consider some helium gas in box. Temperature = 300K,
volume V = 1D“ of, pressure P a ll}5 Mimi. helium gas vacuum a} What is the number N of helium atoms in the box? Now open a small hole in the box and let gas escape into another box that is completely empty,
and has volume [13* ID? m3. b} What is the change in internal energy of the helium gas? Explain in 1 sentence. e] The number of mierostates is proportional to V”, By what faetor have the number of
mjerostates changed? (I) 1What is the change of entropy? Gk — so far so good. We just changed the entropy of the gas by letting it expand. But there is
another way to change the entropy of the gas: we can heatfceol it. e} Same initial conditions as before: Temperature = SUDK, 1volume ”y” = 1133 m3, pressure P = 1i]5
Nituz. To which ﬁnal temperature must we [heaﬂcool] the helium so that its entropy changes by
the same amount as in problem 6'? If you are unsure of your answer for problem d, assume that
the entropy change is 1G"2 JFK. Hint l: for a monontomic gas at constant volume, the heat capacity is c.hr = 3m N kn
Hint 2: d5 = LlQlT
Reminder: in this problem, the trelume is constant! .I‘FF Problem 4 (25 points]. You have just very gently exhaled a helium atom in a room filled with
helium. How long {in seconds) 1will it take for the helium atom, on average, to diffuse to some point on a spherical surface of radius R = l rn surrounding your head? Buckgroundlhintsluseful numbers: Molecular diffusion is a random process: the total
displacement at the mean free path :1, and n (the number of collisions it suffers as it moves
through the displacement x}, are related by x2 = n P. Assume that the effective radius of a helium atom is [1.1 nm. T = SWK. Atomic mass of Helium: 4 gfmol. Volume of 1 mole of helium atoms
in this room: (1.112 mg. If you don't know where to begin, start by deriving an expression for the
mean free path l. ...
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 Spring '08
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